N90-23028 A Methodology For Formulating A Minimal Uncertainty Model For Robust Control System Design and Analysis Christine M. Belcastro MS 489 NASA Langley Research Center Hampton, VA. 23665 B.-C. Chang ME&M Dept. Drexel University Philadelphia, PA. 19104 Robert Fischl ECE Dept. Drexel University Philadelphia, PA. 19104 In the design and analysis of robust control systems for uncertain plants, the technique of formulating what is termed an "M-A model" has become widely accepted and applied in the robust control literature. The "M" represents the transfer function matrix M(s) of the nominal system, and "A" represents an uncertainty matrix acting on M(s). The uncertainty can arise from various sources, such as structured uncertainty from parameter variations or multiple unstructured uncertainties from unmodeled dynamics and other neglected phenomena. In general, A is a block diagonal matrix, and for real parameter variations the diagonal elements are real. As stated in the literature, this structure can always be formed for any linear interconnection of inputs, outputs, transfer functions, parameter variations, and perturbations. However, very little of the literature addresses methods for obtaining this structure, and none of this literature (to the authors' knowledge) addresses a general methodology for obtaining a_ M-A model for a wide class of uncertainty. Since having a A matrix of minimum order would improve the efficiency of structured singular value (or multivariable stability margin) computations, a method of obtaining a minimal M-A model would be useful. This paper presents a generalized method of obtaining a minimal M-A structure for systems with real parameter variations. Robust control theory for both analysis and design has been the subject of a vast amount of research in this decade [1-35]. In particular, robust stability and performance have been emphasized in much of this work, as, for example, in the development of I-I00 control theory [10-15, 19-23]. Moreover, the development of robust control system design and analysis techniques for unstructured [1-9, 13, 19, 21] as well as structured [16-35] plant uncertainty continues to be the subject of much research - particularly the latter. Unstmctm-ed plant uncertainty arises from unmodeled dynamics and other neglected phenomena, and is complex in form. This uncertainty is called "unstructured" because it is represented as a norm-bounded perturbation with no particular assumed structure. Plant uncertainty is called "structured" when there is real parameter uncertainty in the plant model, or when there is unstructured complex uncertainty occurring in the system at multiple points simultaneously. Plant parameter uncertainty can arise from modeling errors (which usually result from assumptions and simplifications made during the modeling process and/or from the unavailability of dynamic data on which the model is based), or from parameter variations that occur during system operation. Robust control design and analysis methods for systems with unstructured uncertainty is accomplished via singular value techniques [1-9, 21 ]. For systems with structured plant uncertainty, however, the structured singular value (SSV) [16-27 ] or multivariable stability margin (MSM) [28-33 ] must be used. In order to compute the SSV or MSM, the system is usually represented in terms of an M-A model. The "M" represents the transfer function matrix M(s) of the nominal system, and "A" represents an uncertainty matrix acting on M(s). In general, A is a block diagonal matrix, and for real parameter uncertainites the diagonal elements are real. As indicated in the literature [17,18,20,28 ], this structure can always be formed for any linear interconnection of inputs, outputs, transfer functions, parameter variations, and perturbations. However, very little of the literature discusses methods for obtaining an M-A model. For unstructured uneertainites, this model is very easy to obtain. However, for real parameter variations, forming an M-A model can he very difficult. In [29 ], De Gaston and Safonov present an M-A model for a third-order transfer function with uncertainty in the location of its two real poles and in its gain factor. Although the given M-A model is easily obtained for this simple example, other examples do not yield such a 355 https://ntrs.nasa.gov/search.jsp?R=19900013712 2020-03-19T22:05:11+00:00Z
Transcript
N90-23028
A Methodology For Formulating A Minimal UncertaintyModel For Robust Control System Design and Analysis
In the design and analysis of robust control systems for uncertain plants, the technique of formulating whatis termed an "M-A model" has become widely accepted and applied in the robust control literature. The "M" representsthe transfer function matrix M(s) of the nominal system, and "A" represents an uncertainty matrix acting on M(s). The
uncertainty can arise from various sources, such as structured uncertainty from parameter variations or multipleunstructured uncertainties from unmodeled dynamics and other neglected phenomena. In general, A is a block diagonalmatrix, and for real parameter variations the diagonal elements are real. As stated in the literature, this structure canalways be formed for any linear interconnection of inputs, outputs, transfer functions, parameter variations, andperturbations. However, very little of the literature addresses methods for obtaining this structure, and none of thisliterature (to the authors' knowledge) addresses a general methodology for obtaining a _ M-A model for a wide
class of uncertainty. Since having a A matrix of minimum order would improve the efficiency of structured singularvalue (or multivariable stability margin) computations, a method of obtaining a minimal M-A model would be useful.This paper presents a generalized method of obtaining a minimal M-A structure for systems with real parametervariations.
Robust control theory for both analysis and design has been the subject of a vast amount of research in thisdecade [1-35]. In particular, robust stability and performance have been emphasized in much of this work, as, for
example, in the development of I-I00 control theory [10-15, 19-23]. Moreover, the development of robust controlsystem design and analysis techniques for unstructured [1-9, 13, 19, 21] as well as structured [16-35] plant uncertaintycontinues to be the subject of much research - particularly the latter. Unstmctm-ed plant uncertainty arises fromunmodeled dynamics and other neglected phenomena, and is complex in form. This uncertainty is called "unstructured"because it is represented as a norm-bounded perturbation with no particular assumed structure. Plant uncertainty iscalled "structured" when there is real parameter uncertainty in the plant model, or when there is unstructured complexuncertainty occurring in the system at multiple points simultaneously. Plant parameter uncertainty can arise frommodeling errors (which usually result from assumptions and simplifications made during the modeling process and/orfrom the unavailability of dynamic data on which the model is based), or from parameter variations that occur duringsystem operation.
Robust control design and analysis methods for systems with unstructured uncertainty is accomplished viasingular value techniques [1-9, 21 ]. For systems with structured plant uncertainty, however, the structured singularvalue (SSV) [16-27 ] or multivariable stability margin (MSM) [28-33 ] must be used. In order to compute theSSV or MSM, the system is usually represented in terms of an M-A model. The "M" represents the transfer function
matrix M(s) of the nominal system, and "A" represents an uncertainty matrix acting on M(s). In general, A is a blockdiagonal matrix, and for real parameter uncertainites the diagonal elements are real. As indicated in the literature
[17,18,20,28 ], this structure can always be formed for any linear interconnection of inputs, outputs, transferfunctions, parameter variations, and perturbations. However, very little of the literature discusses methods forobtaining an M-A model. For unstructured uneertainites, this model is very easy to obtain. However, for realparameter variations, forming an M-A model can he very difficult. In [29 ], De Gaston and Safonov present an M-Amodel for a third-order transfer function with uncertainty in the location of its two real poles and in its gain factor.Although the given M-A model is easily obtained for this simple example, other examples do not yield such a
straight-forward result. A general state model of M(s) for additive real perturbations in the system A matrix (where Ais assumed to be closed-loop) is discussed in [34]. Unfortunately, this model is not general enough for manyexamples, since system uncertainty is restricted to the A matrix and the uncertainty class is restricted to be linear.Morton and McAfoos [26 ] present a general method for obtaining an M-A model for linear (affine) real perturbations
in the system matrices (A,B,C,D) of the open-loop plant state model. In this model, an interconnection matrix P(s) isconstucted first for separating the uncertainties from the nominal plant, and then M(s) is formed by closing thefeedback loop. The M-A model thus formed can be used in performing robustness analysis of a previously determinedcontrol system. If the feedback loop is not closed, Ix-synthesis techniques [19-21] can be applied to the M-A model forrobust control system design. Morton's result essentially reduces to that of [34] when the perturbations occur only inthe A matrix (and the A matrix of [34] is assumed to be open-loop). An algorithm for easily computing M(s) basedon Morton's result is presented in [35]. Although this method of constructing an M-A model is general for linearuncertainties, many realistic problems require a more general class of uncertainties. Furthermore, no consideration isgiven to obtaining a _ M-A model, where "minimal" refers to the dimension of the A (or M) matrix. Since the
M-A model is a nonunique representation, it would be prudent to obtain one of minimal dimension so that thecomplexity of the SSV or MSM computations during robust control system design or analysis could be minimized.However, none of the literature (to the authors' knowledge) addresses the issue of minimality.
This paper presents a methodology for constructing a _ M-A model for systems with real parametricmultilinear uncertainties, where the term "multilinear" is defined as follows:
Definition: A function is multilinear if the functional form is linear (affine) when any variable is allowed to vary
while the others remain fixed. For example, f(a,b,c) = a + ab + bc + abc is a multilinearfunction.
Thus, the allowance of multilinear functions of the uncertain parameters provides a means of handling _ross-terms inthe transfer function coefficients. A procedure is proposed for obtaining this model in state-space form for uncertain
single-input single-output (SISO) systems, given the system transfer function in terms of the uncertain parameters.An extension of this result to multiple-input multiple-output (MIMO) systems will be given in a subsequentpublication. In this development, M(s) will represent the nominal open-loop plant, so that the resulting M-A modelmay be used for robust control system analysis or design. The state-space form used in modeling M(s) is an extensionof Morton's result for real parametric linear (affine) uncertainties [26]. The paper is divided into the followingsections. A formal statement of the problem to be solved in this paper is presented in Section 2, followed by adiscussion of minimality considerations in Section 3. The approach is presented in Section 4, a proposed solution tothe problem is presented in Section 5, and the proposed procedure for finding a minimal M-A model is summarized inSection 6. Several examples demonstraing the proposed solution are given next in Section 7, followed by some
concluding remarks in Section 8.
2. Problem Statement
Given the transfer function of an uncertain system, G(s,5), in either factored or unfactored form, as a
function of the uncertain real parameters, 5, find a minimal M-A model of the form depicted below in Figure 1:
G(s, 5)
q
Y
v
Figure 1. Block diagram of the General M - A Model
such that: 1. The diagonal uncertainty matrix, A, is of minimal dimension.2. The model of the nominal plant, M(s), is in state-space form.
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Themodelmusthandlemultilinearuncertaintyfunctionsinanyor all of the transfer function coefficients. Inorder to construct a minimal M-A model, the dimension of the A matrix must be minimized. Hence, factors whichhave been found to affect the dimension of the M-A model will be discussed next, followed by the approach used in
forming a solution to this problem.
3. biinimalitv Considerations
In constructing an M-A model of an uncertain system, the A matrix can become unnecessarily large due torepeated uncertain parameters on its main diagonal. It is therefore of interest to examine the factors which can causethis repetition, so that the number of repeated uncertain parameters can be minimized. A factor which can be shown tocause unnecessary repetition in the A matrix is the particular realization used in representing the system. Examplescan be constructed which demonstrate this effect, and it appears that a cascade realization (and, in particular, cascadeduncertain real poles and zeros) is a desirable form for obtaining a minimal MoA model. Thus, a general cascade-formrealization will be part of the approach taken in constructing a minimal M-A model. A problem arises, however, inthat some transfer functions have a form which precludes cascading uncertain real poles or zeros, such as:
G(s,_) = bls 2 + b_s + b_(s + 0_)(s + 02)
G(s,8) -- (s + 01)(s+ 02)
S2 + al s + a2
where 01 and 02 are assumed here to be uncertain (and hence a function of S). Cascading the poles and zeros for either
case would result in improper transfer function blocks to be realized. For these cases, it is unavoidable for theminimal A matrix to have repeated uncertain parameters on the main diagonal. However, for each inseparable pole orzero pair it is only necessary to repeat one uncertain parameter. This issue will be addressed in the proposed solution,and a minimal M-A model for the first transfer function above will be given as an example.
Another factor which affects the dimension of the M-A model is the form of the coefficients in the systemtransfer function. If any of the coefficients is a nonlinear function of the uncertain parameters instead of a multilinearfunction (e.g., there are squared uncertain terms in any of the coefficients), then extra dependent uncertain parameters
must be defined in order to represent these terms in a multilinear form. For example, 512 would be represented as
8182 where _i2 = 81, and both 81 and 82 would appear in the A matrix. Thus, for this case, it is again necessary that
the minimal A matrix contain repeated uncertain parameters on its main diagonal. An example illustrating thissituation will be presented later.
These issues are addressed in the proposed solution for constructing a minimal M-A model. The approachtaken in forming this solution is described in the next section.
4. Annroach
Based on the problem def'mition and the minimality considerations outlined above, several issues will be
addressed in forming a solution to the problem of constructing a minimal M-A model given the transfer function of anuncertain system. First, a general cascade-form realization will be found which can be used to obtain a minimal M-Amodel. Second, the minimal A matrix will be determined for any uncertain system such that extra dependent
parameters are assigned to account for inseparable pairs of uncertain real poles or zeros as well as non-multilinear (e.g.,squared) terms. Third, a method of obtaining a state-space realization of M(s) for any uncertain system will be found.Therefore, the proposed approach for constructing a minimal M-A model is given as follows:
1. Obtain a cascade-form realization of the system so that the state-space uncertain model can be written as:
i= Ax + Bu (1)y= Cx + Du
where: A = A + [AA], B = B + [AB], C = C + [AC], D = D + lAD] (2)O O O O
The terms with the "o" subscript (A o, B o, C o, Do) represent the nominal matrix components, and the "A" terms
(AA, AB, AC, AD) represent the uncertain matrix components. To eliminate confusion of the A notation, thediagonal uncertainty matrix, A, of the M-A model will be represented as [A], and the AA, AB, AC, and AD matriceswill be represented as [AA], [AB], [AC], and [AD] wherever clarification is required.
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2. Obtain a minimal M-A model as described in the problem definition and pictured in Figure 1, where:
a. The minimal uncertainty matrix, A, is defined as:
_-d tS1,82, 3..... 8m]-- tSi, ]=d tS] (3)
where: A e Rmxm, 8I 6 Rml, 8D 6 RmD, 8 _ Rm
and: m = the minimal number of uncertain parametersmI = the number of independent parameters given in G(s,8)
mD = the minimal number of dependent (or repeated) parameters
Also: p = [A] q (4)
where: p = the uncertain parameters input to M(s), P ¢ Rmp
q = the uncertain variables output from M(s), q _ Rmq
Since an M-A model is minimal if the dimension of the A matrix, m, is minimal, where m dependson mI and mD, with mI being given and fixed, a formal definition of a minimal M-A model can bestated as follows:
Definition: An M-A model is _ if mD - i.e., the number of dependent (or repeated) parameters in the A matrix -
is minimal (or zero, if possible).
b. The state-space model of the nominal plank M(s), is an extension of Morton's result [26] and hasthe following form:
(5)
where Bxp, Cqx, Dqp, Dqu, and Dyp are constant matrices. Thus, M(s) can also be written in theequivalent shorthand notation defined as follows:
It should be noted that in [26] the D_ matrix was required to be zero. In this paper, however, Dqp isallowed to be nonzero in order to model the multilinear (cross-produc0 uncertain terms.
The results for constructing a minimal M-A model via this approach are presented in the next section.
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5. Proposed Solution
The proposed solution will be presented in two parts: the results for obtaining a cascade-form realization ofthe uncertain system will be summarized fn'st, followed by the results for obtaining the state-space realization of aminimal M-A model.
5.1 Cascade-Form Realization
Given the transfer function of an uncertain system in terms of its uncertain parameters, G(s,8), it is desired torealize the system in a cascade form of first- and second-order subsystems. Thus, if the transfer function is given inunfactored form, the numerator and denominator polynomials must be factored into first- and second-order terms. Thegiven transfer function will then be represented as follows:
G(s,8) = Ky(8) GC(S,8) GR(SJS) Ku(_) (8)
where K u and Ky represent input and output gain terms, respectively, and G R and G C represent the real (first-order) andcomplex (second-order) transfer function components, respectively. Then :
Gc, (s,8) = b3i-2 s2 + b3i-I s + b3is2 + a2i-I s + a2i
(12)
and: k = number of real (first-order) blocks
1 = number of complex (second-order) blocks.
Any or all of these transfer function coefficients may be uncertain. The uncertainty may arise from either thecoefficient itself being uncertain, or from the coefficient being a function of one or more uncertain variables.Therefore, for either case, any of the coefficients may be a function of 8. Furthermore, the uncertain variables mayhave either an additive or multiplicative form:
= eo + r_ , e = eo(1 + _) (13)
The following cascade-form state-space realization of this system is proposed:
G(s) =
m
AR 0
BCCR AC
Ky DC C R gy CCm
BR Ku
EC ER K u
KyECD R K u
(14)
where:
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AR =
AR[ 0 " " "
BR2CR1 A R2 ...
D R B R 3 C R 2 " " "BR 3 2CRI
BR4 DR D R B D CR -..3 2 CR1 R 4 R 3 2
•
BR D R ... DRfR2" ""BR k-IDR k-2""DR 2CR 1 k-1 k-2
The realizations { ARi, BRi, CRi, DRi } and { ACi, BCi, CCi, DCi } represent the ith real (In'st-order) and complex
(second-order) systems GRi(S,5) and Gci(s,8), respectively. Thus, for the real subsystems, i = 1, 2 ..... k, and for the
complex subsystems, i = 1, 2 ..... 1.
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Theresultingcascade-formrealizationoftheuncertainsystem is therefore given from (14) as:
0] B [BRKu]A = Bc CR Ac Bc DR Ku
C = [ Ky Dc CR Ky Cc ] D = Ky Dc DR Ku
(21)
The above model is a general cascade-form realization for any uncertain open-loop SISO transfer function.The model does not, however, handle nonmonic denominator polynomials with uncertain leading coefficients. Thiswould result in fractional (i.e., rational) matrix elements in the realization with uncertain parameters in thedenominator of these elements. For real uncertain poles or zeros, two factors determine whether the real (first-order) orcomplex (second-order) block form should be used. The first is the nature of the uncertainty associated with theseterms, and the second is the form of the transfer function. If the real pole or zero locations are the uncertain parametersand the transfer function form allows these poles or zeros to be separated out, then the real block form should be used.If the transfer function form does not allow this separation, then the complex block form must be used. Furthermore,if there is a p__ of uncertain poles or zeros that cannot be cascaded, then the resulting minimum A matrix will have a
repeating parameter on the main diagonal for each inseparable pole or zero pair. Alternatively, if the coefficients of thesecond-order polynomial associated with the real poles are the uncertain parameters, then the complex block formshould be used. These cases will be illustrated in the Examples section of this paper. The formulation of the
minimal M-A model will be presented next.
5.2 Minimal M-A Model
In formulating the minimal M-6 model, the minimal A matrix must be determined fn'st, followed by thestate-space realization of M(s). Thus, the results for formulating this model will be presented in this order.
5.2.1 Minimal A Matrix
The minimal A matrix is defined as in (3) with:
m -- m I + m D, (22)
where m I is the number of independent uncertain parameters, and m 0 is the number of dependent uncertain parameters
that must be added. The uncertain independent parameters are those defined in G(s,_). However, as discussedpreviously, the dependent uncertain parameters are those independent parameters that must be repeated due to non-multilinear terms in the transfer function coefficients and/or pairs of uncertain real poles or zeros that cannot becascaded. Thus, for A to be minimal, mD (or _iD) should be minimized. It can be shown that if the system transfer
function is formed from a given minimal M-A model of an uncertain system, the coefficients of the numerator anddenominator polynomials will be multilinear functions of the uncertain parameters. Unfortunately, the converse is notnecessarily true in general because of the dependence of the M-A model on the realization used for the plant. If thegeneral cascade-form realization posed in this paper is used, however, the multilinear form of the transfer functioncoefficients can be used to establish that m = m I (i.e., m D = 0), unless there are real uncertain pairs of poles or zeros
that cannot be cascaded. Furthermore, it can be shown that if the coefficients of all the f_tors of the numerator and
denominator polynomials are multilinear functions, then the coefficients of the expanded polynomials will also be
multilinear. However, if there are non-multilinear uncertain terms in the transfer function, then dependent parametersmust be defined (and added to A) to represent the non-multilinear term in a multilinear form. Moreover, if the non-
multilinear term is of the form 5n, then n-1 dependent parameters must be defined. If there are pairs of real uncertainpoles or zeros that cannot be cascaded, then one additional dependent parameter must be added for each pair, and thedependent parameter can be either of the uncertain real parameters in the pair. Therefore, the number m, as determinedby these rules, is the minimal dimension of the A matrix for the uncertainty class considered in this paper. Once thisminimal dimension is determined, the A matrix can be defined as a diagonal matrix, as in (3), with the specifieduncertain parameters on the main diagonal. Examples which illustrate these cases will be presented later in Section 6.
5.2.2 State-Space Realization of M(s)
Once the cascade-form realization has been determined, the system can be modeled as in (l) and (2), where[AA], lAB], [AC], and lAD] are known functions of the uncertain parameters. Since any non-multilinear terms have
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beenredef'med in a multilinear form when the minimal A matrix is determined, these matrices are multilinear functions
of the parameters. In order to obtain a state-space model for M(s) as defined in (5), expressions for these uncertainty
matrices must be determined in terms of the matrices Bxp, Cqx, Dqp, Dqu, and Dyp from the model. Using (4) and
(5), these expressions are determined as follows:
[AA] = Bxp [A ]( I - Dqp [A ])-1 Cq x = Bx p ( I - [A ] Dqp )-1 [A ] Cqx
[AB] = Bxp[A](I-Dqp[A])'lDq u = Bx p(I.[A]Dqp)-l[A]Dqu
The inverse term makes computation of Dqp very difficult. Furthermore, the matrix inversion can cause [AA], [AB],[ACT],and [AD] to have fractional (i.e., rational) elements with uncertain parameters in the denominator, which is notallowed in the uncertainty class being considered. Thus, it is desirable to represent this term in expanded form asfollows:
The second group of terms add in tim cross-terms of the multilinear uncertainty functions. Each term in the series addsa higher-order cross-product term. Since [AA], lAB], [AC], and [AD] are multilinear functions with a f'mite number of
terms, the Dqp matrix can be defined to have a special nilpotent structure such that:
where r is the order of the highest cross-term oecuring in [AA], [AB], [AC], and [AD], i.e.:
r = max ( O A, O B, O C, O D ) (28)
where O A, O B, O C, and O D represent the order of the highest-order cross-product term in [AA], [AB], [AC], and [AD],
respectively. Cross-product term order is defined as:
order(818283...Si) = i-1 (29)
where i = 1, 2 ..... m. Thus, the maximum value ofr is rmax = m-I, where m is the dimension of the A matrix.
The required structure for Dqp to satisfy (26) and (27) is given as follows:
1.) dii = 0; i= 1,2 ..... m
2.) If dij # 0, then for i=1,2 ..... m and j=l,2 ..... m: (30)
a.) dji = 0;
b.) diel,je 1 = 0 or die2,jO 2 = 0 or ... or die(m_l),je(m_l) = 0
where the symbol "e" represents "modulo m" addition. The desired equations can therefore be written as:
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[AA] = Bxp[A]Cqx+Bxp{[A]Dqp + ([A]Dqp)2 + ...
[AB] = Bxp[A]Dqu +Bxp{[A]Dqp + ([A]Dqp)2 + ...
[AC] = Dyp[A]Cqx+Dyp{ [A]Dqp + ([A]Dqp)2 + ...
[AD] = Dyp[A]Dqu+Dyp{ [A]Dqp + ([A]Dqp)2 + ...
( [A ] Dqp )r } [A ] Cqx
( [A ] Dqp) r } [A ] Dqu
([AlDqp)r } [A]Cqx
([a]Dqp) r } [a] Dqu
(31)
where "r" is defined in (28). Since the [AA], [AB], [AC], and [AD] matrices are known for the given system, the
equations in (31) are used to determine Bxp, Cqx, Dqu, Dyp, and Dqp. Once these matrices are obtained, the state-space model of M(s) is found. Hence, a minimal M-A model has been formed.
A procedure which summarizes the necessary steps in obtaining a minimal M-A model using these results ispresented next.
6. Summary of Procedure
The following is a summary of the procedure implied by the above proposed approach for forming a minimalM-A model of a given uncertain system:
i.) Obtain the system transfer function in factored form. The coefficients of each factor should be a multilinear
function of the uncertain parameters. If necessary, define new dependent parameters to represent any non-multilinear terms in a multilinear form.
ii.) Define the number of parameters in the A matrix, m, using (22). In so doing, determine if any new parametersare required to model inseparable uncertain real pole or zero pairs. If there are inseparable real pairs, eitheruncertain parameter in the pair may be repeated.
iii.) Define the minimal A malrix as in (3), using the independent parameters defined in the given transfer functionas well as those defined in steps i.) and ii.) above.
iv.) Obtain a cascade-form realization for the system as a function of the uncertain parameters.
v.) Express the system matrices as in (2).
vi.) Determine the maximum order of cross-product terms, r, in [AA], [AB], [AC], and [AD] as defined by (28) and
(29). Then [AA], [AB], [AC], and [AD] have the form represented in (31), where Dqp has the special(nilpotent) structure summarized by (30).
vii.) Express [AA], [AB], [AC], and [AD] as:
[AA] = [AAo] + [AAI] + [AA2] + ... + [AAr]
[AB] = [ABo] + [ABI] + [AB2] + ... + [&Br]
[AC] = [ACo] + [ACI] + [AC2] + ... + [ACr]
[AD] = [ADo] + [AD1] + [AD2] + ... + [ADr]
(32)
where the subscript i represents the cross-teams of ith order in each uncertainty matrix.
viii.)The Bxp, Cqx ,Dyp, and Dqu matricesarefoundusingtheexpansiondescribedin[26]fortheuncertainty
where the M i matrices are appropriately partitioned. For the case of repeated parameters due to inseparable real
poles or zeros, the Mi malrix associated with the repeated parameter must be no_em. These matrices can be
decomposed into the product of appropriately partitioned column and row malrices as follows:
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MBi ]Mi = M_ [ Mca MDn] (34)
where MBi forms the ith column of Bxp, MD1 i forms the ith column of Dyp, MCi forms the ith column of
Cqx, and MD2 i forms the ith column of Dqu. Thus:
Bxp = [ Ms1 MBz --- MB_ ]
Dyp = [ MI:hl MD12 -'" MDlm]
Cqx v = [ McIT McT ... Mc m]T
D T [ MD21T MI_ T ... MIni T
(35)
ix .) Use the higher-order cross-terms of [AA], [AB], [AC], and [AD], as in (32), to determine the elements of the
Dqp matrix. Begin with the t-n'st-order terms and specify as many elements as possible. Continue with thesecond-order terms, and proceed until all elements of Dqp are specified. Check Dqp to ensure that the required
special structure of (30) and, hence, (26) is satisfied.
x.) Form the minimal M-A model as given in (3), (5) and (6), and as pictured in Figure 1.
It should be noted that the matrices MBi, MCi, MD1 i, and MD2 i, obtained in decomposing the M i matrices
in (34), are not necessarily unique. A method of formalizing this decomposition for computer implementation willnot be addressed in this paper. However, an algorithm is presented in [35] which accomplishes this decomposition fora more restrictive uncertainty class. Some examples will be given next to illustrate these results.
7. F,aam.al_
The following examples illustrate the proposed procedure presented above. Due to space limitations, detailsof each step will not be included. However, the results that are presented for each example should be fairly easilyobtained.
The cascade-form realization of this example is found in a sa-aight-forward manner to be:
m Ill-_1 0 0 0 ] 131
[32-t11_1 -0t2 0 0 ] B =0 0 0 1 _ 0
[_3(_2-0t113 X) 1_4=0t2133 -a2 -al _1133
364
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v
0
o_
B"
0
v
C_ o _o _ o - o o o o
O _ _ C_ P._ C_ 0 C_ C_ C_ 0
Q C_ C_ 0 C_ C_ C_ 0 _ C_ C_
II
C_
C_
C_
C_
C_
o
C_
o
C_
.o.
D
II
I
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o
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v ov
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o
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_-_ 000
_-_ 000
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0000
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I I
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_-_. _
_. =
0
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.._
_"_ _N__N
_N
_" _• _..
i!."P-
N
_'_
[ J
This example illustrates the case of an inseparable uncertain real pole pair. Consider the following system:
G(s,8) = bls 2 + b2s + b3(S + 01)(S + 02)
where: b 1 = bl o+Sb I , b2 = b2o+Sb 2 , b3 = b3o+Sb 3
01 = 01o+801 , 02 = 02o+802
Since the numerator is second order with uncertain coefficients, the uncertain real poles in the denominator cannot beseparated into the real cascade form. The denominator must therefore be expanded, and the complex (second-order)block used in the realization, which is given as follows:
o 1 1, B=[o]-0102 -(01 + 02) _1 1
(b3 - 0102bl) (b2 - (01+02)bl)] , D = [bl]
Since there is one inseparable uncertain pair of poles, either 801 or 802 must be repeated in the A matrix. (It can be
shown that if this is not done, Dqp will not have the required structure and hence the higher-order cross-product termswill not be modeled correctly.) Since there are no non-multilinear terms in any of the transfer function coefficients,m = 6 (i.e., five given independent uncertain parameters plus one dependent parameter for the inseparable pole pair).The resulting A matrix can therefore be deemed as follows:
A = diag [ 801,801,802, 8b 1, 8b 2, 8b 3 ]
where 801 was arbilrarily chosen to be repeated. Using the proposed procedure, the following results are obtained:
This is a second-order system with uncertain complex poles. The uncertainty appears in the teal and imaginarycomponents of the complex poles. The constant coefficient, to2, is not a multilinear function of the uncertain
parameters. Substituting for c and to in the above transfer function yields the following equation:
s2y = u - 2% sy - 0)o2 y - 28 sy - ( 2tooSto+ 8to2 ) y
In forming an M-A model, the problematic term is 5o2 because it is not multilinear. In order tO represent this
equation in multilinear form, the following dependent variable is defined:
The A matrix is given by A = diag [ 8or, 8to, 83 ], where 83 = 8to.
These examples illustrate the proposed procedure for forming a minimal M-A model of an uncertain system.Although all the steps involved in obtaining these results have not been included, the stated results should provide aguide in performing the steps of the proposed procedure. It should be noted that, for ease of hand computation, theexamples included only the simplistic (and less realistic) case in which the coefficients themselves are the uncertainparameters. However, it is emphasized that the proposed procedure does handle the more realistic case in which theuncertain transfer function coefficients are multilinear functions of the uncertain parameters.
367
8. Concludin_ Remarks
This paper has presented a proposed procedure for forming a minimal M-A model of an uncertain systemgiven its transfer function in terms of the uncertain parameters. The uncertainty class considered in this paper allowsthe transfer function coefficients to be multilinear functions of the uncertain parameters, and the uncertainties may arisein the A, B, C, and D matrices of the system model. The proposed procedure involves realizing the system in a
cascade form, determining the minimal A matrix of uncertain parameters, and obtaining a state-space model for thenominal system, M(s). Three examples were given to illustrate the proposed procedure. The first example had elevenindependent uncertain parameters, which arose in the A, B, C, and D matrices of the system realization. The secondexample had uncertain parameters arising in the A, C, and D matrices only. This example illustrated the formulationof a minimal M-A model for a system with inseparable real uncertain poles, and involved repeating an uncertain
parameter in the A matrix. The last example had uncertainty in the A matrix only, and illustrated a method for
handling non-multilinear terms.Further work on the proposed procedure will be to include systems having a nonmonic characteristic
polynomial with an uncertain leading coefficient, as well as systems having an inner feedback loop which may or maynot have uncertainties. The latter case may require a modification in the formulation of the cascade realization.Although the procedure presented in this paper is for SISO systems, an extension to MIMO systems will beforthcoming, and should primarily involve modifying the caseade-form realization. Other areas of future work includedevelopment of a simple means of verifying the minimality of a given M-A model, and development of a method of
reducing a nonminimal M-A model to a minimal form.
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